Quantum information is physical information that is held in the state of a quantum system. The unit of quantum information may be a qubit, a two-level quantum system. In contrast to discrete classical digital states, a two-state quantum system can be in a superposition of the two states at any given time. Unlike classical information, quantum information cannot be read without the state being disturbed by the measurement device. Furthermore, in quantum information, an arbitrary state cannot be cloned.
Coherent states of light, such as those of laser light waveforms, are widely used for communication and sensing applications, so the optimal discrimination of coherent states, that is, the quantum states of light emitted by a laser, has immense practical importance. However, quantum mechanics imposes a fundamental limit on how well different coherent states can be distinguished, even with perfect detectors, and limits such discrimination to have a finite minimum probability of error. While conventional optical detection schemes lead to error rates well above this fundamental limit, an explicit receiver design involving feedback and photon counting that can achieve the minimum probability of error has been proposed. However, this receiver design only applies to a set of two coherent states (the binary case) and the generalization of this proposed design to larger sets of coherent states has proven to be challenging, thereby suggesting that this may be a limitation inherent to proposed linear-optics-based adaptive measurement strategy.
Helstrom (cited reference [1] on page 4 of Appendix I) provided a set of necessary and sufficient conditions on the measurement that yields the minimum average probability of error in discriminating M≧2 distinct quantum states. However, for optical state discrimination, this mathematical specification of measurement operators does not usually translate into an explicit receiver specification realizable using standard optical components, thus leaving a gap between the minimum error probability (the Helstrom limit) and the minimum achievable by conventional measurements, viz., homodyne, heterodyne, and direct detection.
For discriminating two coherent states, Dolinar proposed a receiver that achieves the Helstrom limit exactly for discriminating any two coherent state signals (Cited references [2] and [12] on pages 4 and 5 of Appendix I). This proposed receiver works by applying one of two time-varying optical feedback waveforms to the laser pulse being detected, and instantaneous switching between the two feedback signals at each click event at a shot-noise-limited photon counter. More recently, it has been shown that two coherent states can be optimally distinguished using linear-optical processing followed by adaptive measurements. For discriminating between multiple (M>2) coherent states, there is yet no optical receiver known which achieves the Helstrom limit.
A number of sub-optimal receivers for the M>2 case have been proposed with a common philosophy—that of “slicing” a coherent-state pulse into smaller coherent-states, detecting each slice via photon counting after coherent addition of a local field, and feeding forward the detection outcome to the processing of the next slice, as illustrated in FIG. 1. The Dolinar receiver functions by slicing the coherent state, but instead of compressing all the information content of the slices, it measures each slice individually and feeds information about the measurement forward to the measurement of the other slices. After the final measurement, this receiver is able to determine which state was transmitted with a minimum probability of error allowed by quantum mechanics. The main disadvantages of the Dolinar receiver are that: 1) it can only optimally distinguish between two coherent states, 2) it only optimizes the probability of error and cannot be easily adapted to optimize a different figure of merit, and 3) it results in a classical decision, not quantum states, accordingly, its results cannot be further optimally processed by a quantum device/computer. That is, making a classical decision corresponds to making a measurement, which destroys some information which could be useful for additional quantum processing. Thus, making a classical decision is akin to not fully utilizing the information that is in the quantum states, and thus additional processing after a classical decision may be suboptimal.
A quantum computer makes direct use of quantum mechanical properties, such as superposition and entanglement, to perform operations on data. Contrary to digital computers, which require data to be encoded into binary digits (bits), quantum computers utilize quantum properties to represent data and perform operations on these data. Quantum computers share theoretical similarities with non-deterministic and probabilistic computers, like the ability to be in more than one state simultaneously. A quantum computer maintains a sequence of “qubits,” each of which can represent a one, a zero, or any quantum superposition of these two qubit states. Additionally, a pair of qubits can be in any quantum superposition of 4 states, and three qubits in any superposition of 8.
In general, a quantum computer with n qubits can be in an arbitrary superposition of up to 2n different states simultaneously (contrasting to a classical computer that can only be in one of these 2n states at any one time). A quantum computer operates by setting the qubits in a controlled initial state that represents the hypothesis at hand and by manipulating those qubits with a fixed sequence of quantum logic gates. The calculation may end with measurement of all the states, collapsing each qubit into one of the two pure states, so the outcome can be at most n classical bits of information. Alternatively, the qubits may be stored in a quantum memory for further quantum processing.
Recent research shows that any optical receiver involving the most general coherent-state feedback, passive linear optics and photon counting cannot attain the quantum-limited channel capacity of an optical channel to carry classical information. The fact that such generalizations of Dolinar's optimal binary receiver to larger sets of coherent states has proven to be challenging, and the above result on the limitation of general coherent optical receivers, suggest that this may be a limitation inherent to the linear-optics-based adaptive measurement approach, and that the binary discrimination case is somehow special.